Let $G$ a finite $2$-group. Consider the Group Ring $F_2[G]$ over the field $F_2$ of two elements. Consider the matrix $M$ with $M_{i,j}=p_{i,j}$ for some $p_{i,j}\in F_2[G]$, $i=1,\cdots k$, $j=1,\cdots n$.
Let $V$ be the sub-$F_2[G]$-module of $\oplus^n F_2[G]$ generated by $\langle P_1,\cdots,P_k\rangle$ with $P_i=(p_{i,1},\cdots,p_{i,n})$ for $i=1,\cdots k$.
I am interested in the kernel of the transformation
$$ M:V^n\to V^k. $$
T
